[1701.05228] Recommendation under Capacity Constraints
We have presented a novel approach for providing recommendations that satisfy capacity constraints.We have demonstrated how this generic approach can be applied to three state-of-theart models, PMF [20], GeoMF [15], and BPR [19], so that the items expected usage respects the corresponding capacities

Abstract: In this paper, we investigate the common scenario where every candidate item
for recommendation is characterized by a maximum capacity, i.e., number of
seats in a Point-of-Interest (POI) or size of an item's inventory. Despite the
prevalence of the task of recommending items under capacity constraints in a
variety of settings, to the best of our knowledge, none of the known
recommender methods is designed to respect capacity constraints. To close this
gap, we extend three state-of-the art latent factor recommendation approaches:
probabilistic matrix factorization (PMF), geographical matrix factorization
(GeoMF), and bayesian personalized ranking (BPR), to optimize for both
recommendation accuracy and expected item usage that respects the capacity
constraints. We introduce the useful concepts of user propensity to listen and
item capacity. Our experimental results in real-world datasets, both for the
domain of item recommendation and POI recommendation, highlight the benefit of
our method for the setting of recommendation under capacity constraints.

p_i == (# times user i followed the recommendation)/(# user i - system interactions)

Figure 1: Item capacity scores sorted in decreasing order for the various choices of capacities. (Experimental Results)Figure 2: Location information Scatter plots. (Experimental Results)Figure 3: Effect of capacity trade-off parameter α in range {0, 0.2, 0.4, 0.6, 0.8, 1} on CapMF’s performance in test RMSE, Capacity Loss and Overall Objective. As expected, the higher the α, the higher the RMSE and the lower the Capacity Loss. (Experimental Results)Figure 4: (a), (b): Actual Capacity. (c), (d): Reverse Binning Capacity. Effect of capacity tradeoff parameter α in range {0, 0.2, 0.4, 0.6, 0.8, 1} on CapBPR’s performance in test 0/1 Pairwise Loss, Capacity Loss and Overall Objective. (Performance of CapBPR (Ranking))Figure 5: Training pairwise 0/1 loss for actual capacities. (Performance of CapBPR (Ranking))Figure 6: CapMF on explicit feedback data. (Implicit versus Explicit Feedback)Figure 7: Movielens 100K. Effect of surrogate loss for capacity term. Figure 8: Gowalla. Effect of item capacities on CapMF. Figure 9: Foursquare. Effect of user propensities on CapMF. (Effect of Surrogate Loss for Capacity Term)Figure 10: Average Precision (AP)@{1, 5, 10}. (Effect on Top-N recommendations)Figure 11: Movielens 100K, explicit: Comparison with baselines in WAP, WMCV. (Comparison with Post-Processing Method)›